Reinforcement Learning

Temporal-Difference Learning

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

status: reviewimportance: importantdifficulty 4/5math: graduateread: 20mlive demo

Back to Reinforcement Learning Next: Q-Learning and SARSA (review)

Concept Structure

Temporal-Difference Learning

01Intuition

Start with the picture, metaphor, or geometric mechanism.

02Math

Make the objects explicit and connect them with notation.

03Code

Mirror the equations with runnable implementation details.

04Interactive Demo

Manipulate the mechanism and watch the idea respond.

2prerequisites

1next concepts

3related links

Learner Contract

What this page should let you do.

You are here becauseTemporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

This Reinforcement Learning concept is the current idea: keep the same invariant visible across Intuition, Math, Code, Interactive Demo.

Before thisBellman Equations (review)

2 prerequisites listed; refresh them before leaning on the math or code.

Bellman Equations (review)Monte Carlo Reinforcement Learning (review)

By the end4/4 sections ready | runnable code expected | live demo

Explain the mechanism, trace the main notation, and test one prediction in the live demo.

Do this firstIntuition

Read the intuition before the notation; the math should name a mechanism you already felt.

Then go nextQ-Learning and SARSA (review)

Follow this edge after making one prediction here; the next page should reuse the result, not restart the route.

Test the linkManipulate one control and predict the visible change.Then continue to Q-Learning and SARSA (review)

Claim/source review status

Substantive review recorded

1/1 claims have bounded review metadata; still check caveats and source scope.Metadata-derived; review may be AI-assisted. Not a human certification.

Claims1/1 reviewed

Sources3 cited

Codeattached

Demolive

Reviewed2026-07-03

Updatedpage 2026-07-03

Learning item flow

4/4 sections readyAsk about this Research room

ConceptTemporal-Difference LearningReinforcement Learning EquationTemporal-Difference Learning equation 1Exact equation CodeTemporal-Difference Learning runnable code 1Exact runnable code DemoTemporal-Difference Learning interactive demoInteractive demo ClaimTabular TD(0) policy evaluation updates V(S_t) immediately after obse...Exact claim check SourceReinforcement Learning: An Introduction, Second EditionExact source

ConceptTemporal-Difference LearningReinforcement Learning

3 sources attachedLocal snapshot ready

concept:reinforcement-learning/temporal-difference-learning

Codenearby Predictbefore reveal Roomitem handoff

01IntuitionStart with the picture, metaphor, or geometric mechanism.02MathMake the objects explicit and connect them with notation.03CodeMirror the equations with runnable implementation details.04Interactive DemoManipulate the mechanism and watch the idea respond.

Intuition

Build the mental picture first so the rest of the page has something to attach to.

Section prompt

Monte Carlo RL waits for the episode to end. Bellman backups use a known model to average over possible next states.

Temporal-difference learning sits between those two ideas. It looks at one real transition,

S_t \rightarrow R_{t+1} \rightarrow S_{t+1},

then asks:

If the next state's current value estimate is my best guess of what happens from there, how wrong was my current value estimate?

That question is the learning move. TD does not wait for the complete return $G_t$ . It also does not need the transition probabilities that a Bellman expectation backup needs. It uses the sampled reward plus the current value estimate of the sampled next state.

The price is that the target is partly made from an estimate. That is bootstrapping. The benefit is that the learner can update online, one transition at a time.

Math

Translate the story into symbols, assumptions, and a derivation you can inspect.

Section prompt

Equation 1 $\text{target}_t = R_{t+1}+\gamma V(S_{t+1}),$ Equation 2 $\delta_t = R_{t+1} + \gamma V(S_{t+1}) - V(S_t).$

Assume a fixed policy $\pi$ in a finite discounted MDP. At time $t$ , the agent is in state $S_t$ , takes the action prescribed or sampled under $\pi$ , receives reward $R_{t+1}$ , and lands in $S_{t+1}$ .

TD(0) builds the one-step target

\text{target}_t = R_{t+1}+\gamma V(S_{t+1}),

where $V(S_{t+1})$ is the current value estimate of the sampled next state. If $S_{t+1}$ is terminal, use $V(S_{t+1})=0$ .

The TD error is the gap between that one-step target and the current estimate:

\delta_t = R_{t+1} + \gamma V(S_{t+1}) - V(S_t).

Then TD(0) moves the current state's value estimate by a learning-rate-scaled error:

V(S_t) \leftarrow V(S_t)+\alpha\delta_t.

Equivalently,

V(S_t) \leftarrow (1-\alpha)V(S_t) + \alpha \left[ R_{t+1}+\gamma V(S_{t+1}) \right].

That last form shows the exponential-moving-average view: keep part of the old estimate and mix in the new sampled target.

The comparison is the core idea:

Monte Carlo target: $G_t$ , a complete sampled return.
Bellman expectation target: $\sum_{s'}P(s'\mid s,\pi(s))[r(s,\pi(s),s')+\gamma V(s')]$ , a model average.
TD(0) target: $R_{t+1}+\gamma V(S_{t+1})$ , one sampled step plus a bootstrapped next value.

Code

Keep the implementation aligned with the notation so the algorithm is legible.

Section prompt

Runnable code 1gamma = 0.90 alpha = 0.40 values = {"Start": 1.20, "Practice": 1.80, "Confused": 0.80, "Maste...python

gamma = 0.90
alpha = 0.40
values = {"Start": 1.20, "Practice": 1.80, "Confused": 0.80, "Mastered": 0.0}

stream = [
    ("Start", "drill", 1.0, "Practice"),
    ("Practice", "repeat", -0.6, "Confused"),
    ("Confused", "ask", 0.7, "Practice"),
]

for state, action, reward, next_state in stream:
    bootstrap = 0.0 if next_state == "Mastered" else values[next_state]
    target = reward + gamma * bootstrap
    td_error = target - values[state]
    update = alpha * td_error
    values[state] += update
    print(
        state,
        action,
        "target", round(target, 3),
        "error", round(td_error, 3),
        "new", round(values[state], 3),
    )

Notice the online dependence: the third line uses the already-updated Practice value. TD learning is not just a formula applied to a frozen table; it is a stream of value repairs.

Interactive Demo

Use direct manipulation to connect the explanation to a moving system.

Section prompt

Use the Temporal-Difference Update Lab to predict which state value changes most after a short sampled transition stream.

Choose a stream, inspect the current values and the sampled transitions, then commit to the state whose estimate should change most. After reveal, the lab shows the TD target, TD error, learning-rate-scaled update, and before/after value table for every transition.

The comparison strip keeps the boundary clear: Monte Carlo waits for a complete return, a Bellman backup needs model probabilities, and TD updates now from one sampled transition plus a bootstrapped next value.

Live Concept Demo

Explore Temporal-Difference Learning

The stage is code-native and interactive. Use it to test the explanation against the mechanism.

difficulty 4/5graduatecode-aligned

Demo Prediction Checkpoint

Manipulate one control and predict the visible change.

Commit to what Temporal-Difference Learning should make visible before reading the result.

After The First Pass

Turn the concept into an inspected object.

Once the invariant is visible in the intuition, math, code, and demo, use these panels to inspect the mechanism visually, check source support, practice the idea, and attach a grounded research question.

Mechanism Storyboard

See the idea move before the page explains it

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

Prediction open01 / Intuition

Prediction lens

Start with the picture, metaphor, or geometric mechanism.

Commit first

Before reading further, choose the kind of change Temporal-Difference Learning should make visible.

Visual Inquiry

Make the image answer a mathematical question

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

4/4 stages readyLive demo connected

Visual cueWhich visible object should carry the first intuition?

Inspection depth2/4

Prediction

Which visible object should carry the first intuition?

Commit first

Pick the cue that should make Temporal-Difference Learning easier to reason about before the page gives the answer.

Source Grounding

Canonical references for the mechanism on this page.

book · 2018Reinforcement Learning: An Introduction, Second EditionRichard S. Sutton and Andrew G. Barto

Canonical source for TD learning as a combination of Monte Carlo experience and dynamic-programming bootstrapping, TD(0), one-step TD targets, and TD error.

Open source

course-notes · 2026Stanford CS234: Lecture 3, Model-Free Policy EvaluationStanford CS234

Graduate RL source for TD(0) targets, TD error, immediate update after an observed transition, and MC-vs-TD bias/variance framing.

Open source

course-notes · 2026UC Berkeley CS188: Model-Free LearningUC Berkeley CS188

Course-note source for passive model-free learning, temporal-difference sample values, exponential-moving-average update, and the Q-learning downstream bridge.

Open source

Claim Review

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

Status1 substantive review recorded

Claims without a substantive review badge still need exact source-support review.

Sources3 references

sutton-barto-2018-rl, stanford-cs234-td-learning, berkeley-cs188-model-free-learning

Local checks4 local checks

Use equations, runnable code, and demos to check whether the source support is operational.

Substantively reviewedTabular TD(0) policy evaluation updates V(S_t) immediately after observing one transition by moving V(S_t) toward the one-step target R_{t+1} + gamma V(S_{t+1}); the bracketed difference between that target and the current estimate is the TD error.Claim metadata: source checked

The sources support finite tabular TD(0) policy evaluation as a model-free, one-step sample update that bootstraps from the current next-state value estimate instead of waiting for a complete Monte Carlo return or averaging over a known transition model.

Sources: Reinforcement Learning: An Introduction, Second Edition, Stanford CS234: Lecture 3, Model-Free Policy Evaluation, UC Berkeley CS188: Model-Free LearningFinite tabular on-policy teaching MDP only; excludes TD control, SARSA, Q-learning, n-step returns, eligibility traces, off-policy corrections, convergence proofs, stochastic-approximation schedules, continuing-task average reward, and function approximation.A bounded review summary is present; still check caveats and exact reference scope.

Checked Sutton/Barto Chapter 6 for TD as the combination of Monte Carlo experience and DP bootstrapping, the TD(0) update, one-step TD target, and TD error; checked Stanford CS234 Lecture 3 for target/error/update framing and immediate update after an observed transition; checked Berkeley CS188 model-free learning notes for passive TD learning as a sample-value exponential update. GPT Pro publication critique remains pending because 127.0.0.1:51672 is unavailable.

Reviewer: codex-local-source-review; reviewed 2026-07-03

source-span-sutton-barto-2018-rl source-span-stanford-cs234-td-learning source-span-berkeley-cs188-model-free-learning Equation 1 Equation 2 Runnable code 1 Interactive demo

Source support candidates

book 2018Reinforcement Learning: An Introduction, Second Edition

Canonical source for TD learning as a combination of Monte Carlo experience and dynamic-programming bootstrapping, TD(0), one-step TD targets, and TD error.

course-notes 2026Stanford CS234: Lecture 3, Model-Free Policy Evaluation

Graduate RL source for TD(0) targets, TD error, immediate update after an observed transition, and MC-vs-TD bias/variance framing.

course-notes 2026UC Berkeley CS188: Model-Free Learning

Course-note source for passive model-free learning, temporal-difference sample values, exponential-moving-average update, and the Q-learning downstream bridge.

Mechanism checks

Equation 1

\text{target}_t = R_{t+1}+\gamma V(S_{t+1}),

Equation 2

\delta_t = R_{t+1} + \gamma V(S_{t+1}) - V(S_t).

Runnable code 1gamma = 0.90 alpha = 0.40 values = {"Start": 1.20, "Practice": 1.80, "Confused": 0.80, "Maste...Demo stateLive mechanism probe

Practice Loop

Try the idea before it explains itself

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

Readiness0/3 checks ready

Predict

Before touching the demo, predict one visible change that should happen in Temporal-Difference Learning.

Hint 1

Reveal when your model needs a nudge.

Hint 2

Reveal when your model needs a nudge.

Hint 3

Reveal when your model needs a nudge.

Your answer canvas

Local checks

Claim

A concrete answer is on the canvas.

Mechanism

The answer names why the claim should hold.

Bridge

It touches the page context or a neighboring idea.

Misconception check

Grounded research drawerClose

ConceptTemporal-Difference LearningReinforcement Learning

Runnable code comparisonTemporal-Difference Learning runnable code 1gamma = 0.90 alpha = 0.40 values = {"Start": 1.20, "Practice": 1.80, "Confused": 0.80, "Maste...Prediction before revealTemporal-Difference Learning interactive demoManipulate one control and predict the visible change.

Grounded room questionWhat is the smallest example that makes Temporal-Difference Learning click without losing the math?Local snapshot ready

Research Room

Attach the question to a claim, equation, code, or demo

Pick the concept, equation, source, runnable code, claim, misconception, or demo state before asking for help. The handoff keeps that page item in context.

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conceptReinforcement Learning

Temporal-Difference Learning

Attached question

What is the smallest example that makes Temporal-Difference Learning click without losing the math?

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Evidence to inspect

References to inspect: attached references on this page.
Definition, prerequisite, and contrast concept links
The equation or runnable code that makes the concept operational
One demo state that shows the invariant instead of a slogan

What would resolve this

The learner can state the mechanism in their own words
The learner can name the prerequisite that would repair confusion
The learner can predict how the mechanism changes under one perturbation

Grounded AI handoff

I am working in Continuous Function's research reading room. Object: concept - Temporal-Difference Learning Selected item key: recorded for copy. Context: Reinforcement Learning Page anchor: recorded for copy. Open question: What is the smallest example that makes Temporal-Difference Learning click without losing the math? Evidence to inspect: - References to inspect: attached references on this page. - Definition, prerequisite, and contrast concept links - The equation or runnable code that makes the concept operational - One demo state that shows the invariant instead of a slogan What would resolve this: - The learner can state the mechanism in their own words - The learner can name the prerequisite that would repair confusion - The learner can predict how the mechanism changes under one perturbation Answer as a careful research tutor: stay source-grounded, separate verified evidence from assumptions, name the relevant math objects, and end with one next action.

View it in context

concept/concept-notebook/reinforcement-learning/temporal-difference-learning
concept:reinforcement-learning/temporal-difference-learning

Learning Map

Before / Now / Try / Next

BeforeBellman Equations (review)

NowIntuition -> Math -> Code -> Demo

TryManipulate one control and predict the visible change.

NextQ-Learning and SARSA (review)

Intuitionready
Mathready
Codeready
Interactive Demoready

AI Companion

Ask beside the selected object

Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target.

Your question

GoalComfortStyleStuck on

Context prompt

You are my AI learning companion for Continuous Function. Current context: Reinforcement Learning concept. Learning surface: Temporal-Difference Learning. What this page says: Temporal-difference learning updates a state value after one sampled transition by comparing the old value with a reward-plus-next-value target. Current section: Intuition, math, code, and interactive demo. Suggested next step: Manipulate one control and predict the visible change.. Learner goal: Understand the idea. Learner comfort level: New to this. Preferred explanation style: Visual first. Task: Explain the central idea in plain language, then restate it with the exact math objects from the page. Answer in a way that helps me learn: ask one clarifying question only if needed, use intuition before notation, and end with one thing I should try on the page.

Domain

Reinforcement Learning

reinforcement-learningtemporal-difference-learningtd-errorbootstrappingpolicy-evaluation

Prerequisites

Bellman Equations (review)Monte Carlo Reinforcement Learning (review)

Leads To

Q-Learning and SARSA (review)

Dynamic Programming and Value Iteration (review)Policy Iteration (review)Markov Decision Process Formalism (review)

Within this domain